Finding The Width: Rectangle Problem Explained

by Alex Johnson 47 views

Let's dive into a classic geometry problem! We're given a rectangle where the length is twice its width, and its perimeter is 48 cm. Our mission? To find the width of this rectangle. It's a great exercise in applying basic algebraic principles and understanding geometric properties. This guide will walk you through the solution step-by-step, making sure you grasp the concepts clearly.

Understanding the Problem

First, let's break down the information. We know a few key things:

  • The Shape: We're dealing with a rectangle. Remember, a rectangle has four sides, with opposite sides being equal in length, and all angles are right angles (90 degrees).
  • Relationship between Length and Width: The problem states the length is twice the width. We can represent this mathematically.
  • Perimeter: The perimeter of the rectangle is 48 cm. The perimeter is the total distance around the outside of the rectangle.

To solve this, we'll use a bit of algebra and our understanding of what a rectangle is. The core of solving this kind of problem is translating the words into mathematical equations. Don't worry, it's simpler than it sounds! We'll define variables, set up an equation, and solve for the unknown (the width).

Let's consider the problem logically. We know the perimeter, which means we have a total length to work with. We also know how the length and width relate to each other. Putting these two pieces of information together allows us to find the individual dimensions. Think of it like a puzzle. Each piece of information is a piece of the puzzle that we can use to reveal the answer.

Setting Up the Equation

Let's define our variables:

  • Let w represent the width of the rectangle.
  • Since the length (l) is twice the width, we can represent the length as 2w. It's important to translate the verbal information into mathematical expressions. This is where the magic of algebra starts to shine. It helps us visualize the relationship between different quantities.

Now, let's remember the formula for the perimeter of a rectangle:

  • Perimeter = 2 * (length + width) or P = 2(l + w)

Substitute the values we know:

  • P = 48 cm
  • l = 2w
  • w = w

So our equation becomes:

  • 48 = 2(2w + w)

This single equation represents all the information we have about our rectangle. Now, we just need to solve it to find the width.

Solving for the Width

Now, let's simplify and solve the equation step by step:

  1. Simplify inside the parentheses: 2w + w = 3w. So our equation becomes 48 = 2(3w).
  2. Multiply: 2 * 3w = 6w. Now, we have 48 = 6w.
  3. Isolate w (the width): To get w by itself, we need to divide both sides of the equation by 6. 48 / 6 = 8. Thus, w = 8.

Therefore, the width of the rectangle is 8 cm. This process is systematic: we use the perimeter formula, which provides the overall size, and the relationship between length and width to find the individual components. The final step is to interpret our result.

Understanding the Solution

We've found that the width (w) is 8 cm. Now, let's find the length to check if our answer makes sense. Since the length is twice the width, the length would be 2 * 8 cm = 16 cm. Now, let's verify if the perimeter is indeed 48 cm by using the formula P = 2(l + w):

  • P = 2(16 cm + 8 cm) = 2(24 cm) = 48 cm

Our solution fits perfectly! The width is 8 cm, the length is 16 cm, and the perimeter is indeed 48 cm. Understanding this process, along with the correct application of the formula, is the key to solving geometry problems efficiently. Always check your answers to ensure the solution makes sense within the context of the original problem.

The Correct Answer

Looking back at our multiple-choice options, the correct answer is:

  • A. 8 cm

This simple problem beautifully illustrates how math concepts build on each other. We use our knowledge of rectangles, the concept of perimeter, and some basic algebra to find the solution. Each step is important, and each concept has its place.

This problem-solving strategy can be applied to many other geometry problems. The ability to translate words into mathematical expressions is essential, and with a little practice, you can master these types of questions. Remember to always define your variables clearly, set up your equations correctly, and then solve for the unknown. Always check your answer to make sure it is accurate!

Tips for Similar Problems

  1. Draw a Diagram: Sketching the rectangle and labeling the sides can make it easier to visualize the problem. Visualization is an important part of the learning process. It enables us to see the abstract concepts in a concrete and tangible way. It's often easier to solve geometry problems if you have a visual representation to refer to.
  2. Write Down the Formulas: Always start by writing down the relevant formulas (perimeter of a rectangle, etc.). Having these formulas at your fingertips is very useful.
  3. Double-Check Your Work: After finding your answer, always plug it back into the original problem to ensure it makes sense. This helps you catch any calculation errors.
  4. Practice Regularly: The more you practice, the more comfortable you'll become with these types of problems. Practice makes perfect. Regularly solving geometry problems will help reinforce your understanding.

By following these steps, you'll be well on your way to solving geometry problems with confidence. Geometry is all about understanding shapes, their properties, and how they relate to each other. The more you work with these shapes and apply mathematical concepts, the more you will understand geometry.

Conclusion

In this problem, we successfully calculated the width of a rectangle given its perimeter and the relationship between its length and width. We’ve covered everything from setting up the equations to arriving at the final answer. This highlights the importance of translating word problems into mathematical equations, a critical skill in problem-solving. By understanding the properties of a rectangle, the concept of perimeter, and basic algebra, you can tackle similar problems with ease. Keep practicing, and you'll find yourself getting better and more confident at each step.

This journey underscores that mathematical problem-solving is not just about calculations, but also about understanding and applying concepts. We started with the question, broke down the problem, applied the right formulas, and solved it systematically. Remember that each concept, when understood and applied, adds to your growing collection of problem-solving skills.

For further exploration, you can check out this trusted resource: Khan Academy - Perimeter of a rectangle